Strominger-Yau-Zaslow conjecture for Calabi-Yau hypersurfaces in the Fermat family (Q2161433)

The article discusses the Strominger-Yau-Zaslow (SYZ) conjecture, which involves the degeneration of \(n\)-dimensional polarized Calabi-Yau manifolds to the large complex structure limit. The article explores the history of research on the SYZ conjecture and the difficulty of the metric problem, including Boucksom et al.'s proposal to prove the SYZ conjecture through the convergence of Calabi-Yau metrics in a non-Archimedean (NA) space. The article then focuses on the Fermat family of projective hypersurfaces and presents new results concerning the convergence of Calabi-Yau metrics on these manifolds. The most striking aspect of their result is the following. \textbf{Theorem}. For the Fermat family, consider the Calabi-Yau metrics on \(X_s\) in the polarisation class \(s^{-1}[\Delta]\), where \([\Delta]\) is a fixed Kähler class on \(\mathbb{CP}^{n+1}\) restricted to \(X_s\). Then, for a subsequence of \(X_s\) as \(s\to\infty\), there exists a special Lagrangian \(T^n\)-fibration on the generic region \(U_s\subset X_s\) such that \[ \frac{\mathrm{Vol}(U_s)}{\mathrm{Vol}(X_s)}\to 1 \text{ as } s\to \infty. \]